Synoptic Meteorology II: The Q-Vector Form of the Omega Equation. 3-5 March 2015

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1 Snoic Meeorolo II: he Q-Vecor Form o he Omea Eqaion 3-5 March 15 eadins: Secion.3 o Midlaide Snoic Meeorolo. Moiaion: Wh Do We Need Anoher Omea Eqaion? he qasi-eosrohic omea eqaion is an ecellen dianosic ool and has sered as he one o he ondaions or inrodcor snoic-scale weaher analsis or i or more ears. Howeer, i is no an inallible ool. aher, i has wo rimar shorcomins: he wo rimar orcin erms in he qasi-eosrohic omea eqaion, he dierenial eosrohic orici adecion and Lalacian o he oenial emerare adecion erms, oen hae dieren sins rom one anoher. Wiho comin he acal manide o each erm, i is diicl i no imossible o assess which one is larer (and hs eers a rimar conrol on he snoic-scale erical moion). he qasi-eosrohic omea eqaion is sensiie o he reerence rame saionar or moin wih he low in which i is comed. In oher words, dieren resls are obained i he reerence rame is chaned, een i he meeoroloical eares are he same! his is no a rimar concern or s in his class, howeer, as we are rimaril considerin saionar reerence rames (e.., as deiced on sandard weaher chars). hese shorcomins moiae a desire o obain a new eqaion or snoic-scale erical moions ha does no hae hese roblems. his eqaion, known as he Q-ecor orm o he qasi-eosrohic omea eqaion, is deried below. Obainin he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion o obain he Q-ecor orm o he qasi-eosrohic omea eqaion, raher han sar wih he qasi-eosrohic orici and hermodnamic eqaions, we will sar wih he qasieosrohic horizonal momenm and hermodnamic eqaions. For simlici, we will assme ha he Coriolis arameer is consan, i.e.,, sch ha all erms inolin β, he meridional ariabili in, are zero. We will also nelec diabaic heain. hs, or basic eqaion se is ien b: a (1) a () he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 1

2 he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae S ω (3) Firs, ind */ o (1): a (4) Ne, ind / */ o (3): S ω (5) We need o ilize he chain rle on he second and hird arial deriaies on he le-hand sides o boh (4) and (5). Where alicable, we also wish o comme (or chane he order o) he arial deriaies; noe ha his alies all erms on he le-hand sides o (4) and (5). his enables s o wrie as man erms as ossible in erms o / and /. he reasons or doin so will become clear shorl. Al he chain rle, comme he arial deriaies as alicable, and come (4) (5), roin like erms where ossible o obain: S a ω (6) For rher simliicaion, recall he hermal wind relaionshi, inrodced in an earlier lecre: (7a) (7b) Noe ha in (7), we hae sbsied or ien ha we are crrenl assmin ha consan. B alicaion o (7b) ino (6), he irs se o erms on he rih-hand side o (6) oes awa. his is wh we rewroe he erms o (4) and (5) as described aboe o make se o he hermal wind relaionshi o simli he resl! Likewise, we can al boh (7a) and (7b) o rewrie he second se o erms on he rih-hand side o (6) in erms o /. Doin so, we obain:

3 he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 3 S a ω (8) ecall ha, b deiniion, or consan, he dierence o he eosrohic wind is zero. his is eqialen o sain ha: (9) I we sbsie (9) ino he irs erm on he rih-hand side o (8), we ind ha i is eqal o he second erm on he rih-hand side o (8). Likewise, i is aaren ha he las wo erms on he rih-hand side o (8) are eqialen. hs, S a ω (1) Finall, noin ha S σ/, i we sbsie or S, diide (1) b, and hen mlil b, we obain: a ω σ (11) In (11), noe ha: Q (1) Sch ha: Q a ω σ (13) Now, we wish o ind */ o (): a (14) Ne, ind / */ o (3):

4 he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 4 S ω (15) As beore, al he chain rle, comme he arial deriaies as alicable, and come (14) (15), roin like erms where ossible o obain: S a ω (16) B alicaion o (7a), he irs se o erms on he rih-hand side o (16) oes awa. Likewise, alicaion o (7a) and (7b) allows s o rewrie he second se o erms on he rih-hand side o (16) in erms o /. Doin so, we obain: S a ω (17) B alicaion o (9), he irs erm on he rih-hand side o (17) is eqal o he second erm on he rih-hand side o (17). Noe also ha he las wo erms on he rih-hand side o (17) are also eqal o one anoher. Simliin (17) wih his while also sbsiin or S, diidin b, and mlilin b, we obain: a ω σ (18) In (18), noe ha: Q 1 (19) Sch ha: 1 Q a ω σ () Eqaions (13) and () conain orcin erms relaed o boh he erical moion ω and aeosrohic wind a. We wish o eliminae he laer. We do so b comin / o ()

5 / o (13). I we do so, commin he deriaies on he aeosrohic wind erms in so doin, we obain: ω ω a a Q1 Q σ σ (1) he las erm on he le-hand side o (1), reresenin he dierence o he aeosrohic wind, can be sbsied or b makin se o he orm o he conini eqaion alicable in he qasi-eosrohic ssem, Sbsiin () ino (1) and simliin he eqaion, we obain: ω a () ω σ ω Q (3) Where Q (Q 1, Q ), wih Q 1 and Q as deined in (19) and (1), reseciel. Basic Inerreaion and Alicaion o he Q-Vecor Eqaion Eqaion (3) ies he Q-ecor orm o he qasi-eosrohic omea eqaion. he qasieosrohic erical moion is de enirel o Q-ecor dierence (where ( ) is he dierence oeraor), which we can readil come and/or esimae. Unlike wih he relar orm o he qasi-eosrohic omea eqaion, here are no mlile orcin erms ha ma conlic wih one anoher, a siniican adanae! Likewise, here are no arial deriaies wih resec o ressre, meanin ha we onl need daa on one isobaric leel o dianose he erical moion anoher adanae! As wih he relar orm o he qasi-eosrohic omea eqaion, we al his eqaion or he dianosis o middle roosheric erical moions. Noe, howeer, ha we sill do no acall sole or he erical moion, ien he second-order arial deriaie oeraors on he lehand side o (3) ha reqire ieraie mehods o sole, as reiosl described. he basic inerreaion o (3) is sraihorward. ecallin ha ω ω, ω Q. hs, Snoic-scale ascen (ω < ) is ond where here is Q-ecor conerence ( Q < ). Snoic-scale descen (ω > ) is ond where here is Q-ecor dierence ( Q > ). he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 5

6 Wiho acall comin Q, is comonens (namel, he horizonal arial deriaies o and ), and is dierence, how can we bes esimae Q and is dierence rom a weaher ma? We ilize a minor coordinae ransormaion, ino a coordinae ssem akin o a naral coordinae ssem, in order o aid in esimain Q and is dierence. Le s deine he -ais o be alon, or arallel o, an isoherm, wih warm air o he rih o he osiie -ais. he -ais is deined erendiclar o he -ais. An idealized schemaic o his is roided in Fire 1 below. Fire 1. Idealized deicion o he coordinae ransormaion described in he e aboe. his is akin o lacin he -ais alon he direcion in which he hermal wind blows, alhoh i shold be noed ha we are onl considerin emerare on one isobaric leel and no a laermean emerare. In his coordinae ssem, /, or he chane in emerare alon he isoherm, is inherenl zero. hs, he erms in he deiniions o Q 1 and Q in (19) and (1) aboe, reseciel, ha inole / are. As a resl, Q becomes: Q i j (4) I we make se o (9), we can rewrie he second erm o (4), sch ha: Q i j (5) And, b ecor ideni, he erm in he arenheses o (5) can be rewrien, sch ha: he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 6

7 Q k (6) hs, o ealae Q, we irs wan o ind he ecor chane in alon he isoherm. he k () oeraor, eomericall, is manies like he rih hand rle and siniies a 9 clockwise roaion o he js-deermined chane ecor. Ne, we mlil his ecor b he manide o /, or he manide o he emerare radien. I is no elicil necessar o mlil b / since, on an isobaric srace, boh and are consan in sace. I we ollow his rocedre a seeral locaions on a weaher ma, we can esimae he dierence o Q and, hs, esimae a leas he sin o he erical moion. Le s now consider hree eamles o sch alicaion. As we do so, we will demonsrae how he same answer is ien b he Q-ecor analsis as wold be obained rom he relar orm o he qasi-eosrohic omea eqaion. Eamle 1: Idealized roh/ide Paern Fire. Q ecors (solid re arrows) or an idealized roh/ride aern. Isoherms are deiced in black dashed lines, wih cold air o he norh, and sreamlines deicin he eosrohic low are deiced in solid reen lines. In he icini o he areas o hih ressre, he eosrohic wind is norherl alon he osiie -ais (o he eas) and soherl alon he neaie -ais (o he wes). In he icini o he area o low ressre, he eosrohic wind is soherl alon he osiie -ais (o he eas) and norherl alon he neaie -ais (o he wes). In each case, we wan o sbrac he eosrohic wind ecor alon he neaie -ais rom he eosrohic wind ecor alon he osiie -ais. o do so, i is hell o recall rinciles o ecor sbracion. o sbrac wo ecors, ake he ecor bein sbraced, li i 18, and add he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 7

8 i o he irs ecor. Vecors are added b lacin he oriin o he second ecor a he i/end o he irs ecor, hen b drawin a new ecor rom he oriin o he irs ecor o he i/end o he second ecor. his is deiced in Fire 3 below or boh siaions described aboe. Fire 3. Illsraion o he ecor sbracion oeraions described in he e aboe. Aer sbracin he ecors, alicaion o he k () oeraor necessiaes roain he new ecor 9 o he rih. his resls in a ecor oinin rom eas o wes wih he areas o hih ressre and a ecor oinin rom wes o eas wih he area o low ressre. he recise lenh o each ecor can hen be deermined b mlilin he ecor b he manide o he emerare radien. he aern o erical moion can hs be deermined b ealain he dierence o he Q ecors. o he wes o he area o low ressre, here is dierence o he Q ecor. his imlies descen. o he eas o he area o low ressre, here is conerence o he Q ecor. his imlies ascen. For a sani check, le s comare his ealaion wih ha which can be obained rom he qasi-eosrohic omea eqaion. o he wes o he area o low ressre, here is cold eosrohic emerare adecion, icall associaed wih descen. Conersel, o he eas o he area o low ressre, here is warm eosrohic emerare adecion, icall associaed wih ascen. Boh o hese indins are consisen wih or Q ecor-based inerreaion. Since he manide o he eosrohic relaie orici ζ is maimized a he base o rohs and ae o rides, we can iner cclonic eosrohic relaie orici adecion o he eas o he area o low ressre and anicclonic eosrohic relaie orici adecion o he wes he area o low ressre. I we resme ha he eosrohic relaie orici adecion is relaiel small in he lower rooshere, his imlies cclonic eosrohic relaie orici adecion increasin wih heih o he eas o he area o low ressre and anicclonic eosrohic relaie orici adecion increasin wih heih o he wes he area o low ressre. his imlies ascen eas and descen wes o he area o low ressre, aain consisen wih or Q ecor-based inerreaion. he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 8

9 Eamle : Idealized roh/ide Paern Wih No emerare Adecion Fire 4. Q ecors (solid re arrows) or an idealized roh/ride aern. Isoherms are deiced in re dashed lines, wih cold air o he norh, and sreamlines deicin he eosrohic low are deiced in solid black lines. In man was, his eamle is similar o he one resened aboe. Howeer, in his case, he isoherms are arallel o he eosrohic wind which, b he deiniion o he eosrohic wind, means ha he isoherms are arallel o he conors o consan eooenial heih. In he ollowin, as beore, eas reers o he osiie -ais alon an isoherm while wes reers o he neaie -ais alon an isoherm. In he base o he roh, he eosrohic wind is rom he sohwes o he eas and rom he norhwes o he wes. Sbracin he laer ecor rom he ormer resls in a ecor oinin rom soh o norh. Alin he k () oeraor roaes his ecor 9 o he rih, sch ha he Q ecor oins rom wes o eas. In he ae o he rides, he eosrohic wind is rom he norhwes o he eas and rom he sohwes o he wes. Sbracin he laer ecor rom he ormer resls in a ecor oinin rom norh o soh. Alin he k () oeraor roaes his ecor 9 o he rih, sch ha he Q ecor oins rom eas o wes. he recise manide o each Q ecor can be obained b mlilin each b he manide o he emerare radien. In Fire 4, i is clear ha Q ecors conere o he eas o he roh and diere o he wes. As beore, his siniies ascen and descen, reseciel. Le s aain inerre he scenario deiced in Fire 4 in erms o he qasi-eosrohic omea eqaion. Wih no eosrohic emerare adecion, orcin is eclsiel de o dierenial eosrohic orici adecion. he aern o dierenial eosrohic orici adecion is idenical o ha described in or irs eamle or he same hsical reasons: cclonic eosrohic relaie orici adecion increasin wih heih o he eas o he roh and he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 9

10 anicclonic eosrohic relaie orici adecion increasin wih heih o he wes. As beore, his imlies ascen and descen, reseciel, consisen wih he Q ecor inerreaion. Eamle 3: Conlen Flow in he Enrance eion o a Je Sreak Fire 5. Orienaion o Q ecors (solid re arrows) or conlen low (inerred rom he reen sreamlines obained rom he eosrohic low) associaed wih a weserl je sreak. Isoherms are deiced b he dashed black lines wih cold air o he norh. In he conlen low scenario deiced aboe, he eosrohic wind acceleraes (becomes larer) o he eas. hs, alon an isoherm, he manide o he eosrohic wind rimaril weserl is alwas larer o he eas. Vecor sbracion resls in a relaiel shor ecor oinin rom wes o eas. Alin he k () oeraor o his ecor roaes i o he rih b 9, sch ha i oins rom norh o soh, as deiced in Fire 5 aboe. As deiced in Fire 5, here is no elici Q ecor conerence or dierence. hs, wha do he Q ecors look like rher o he norh and soh? Visall, we can see ha alon he isoherms, he sreamlines are no mch dieren in direcion or in ackin/ihness. his imlies no meaninl chane in he direcion or manide o he eosrohic wind alon hese isoherms, rher imlin ha he manide o he Q ecors is relaiel small. hs, Q ecors are dieren o he norh, in he colder air, and coneren o he soh, in he warmer air. hs, we see ascen o he soh and descen o he norh. We will reisi his and relaed conces when we sd je sreaks laer in his corse. Aain, we wish o conirm or ealaion b ilizin armens associaed wih he qasieosrohic omea eqaion. here is imlied warm eosrohic emerare adecion (and hs imlied ascen) o he soh and cold eosrohic emerare adecion (and hs imlied descen) o he norh. he inerreaion in erms o dierenial eosrohic orici adecion is slihl more nanced. he sreamlines iml he resence o a roh o he sohwes and a he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 1

11 ride o he norhwes. his imlies cclonic eosrohic relaie orici adecion on he sohern side o he je sreak and anicclonic eosrohic relaie orici adecion on he norhern side o he je sreak. Aain resmin ha eosrohic relaie orici adecion is weak near he srace, his aern imlies ascen o he soh and descen o he norh. Once aain, his is consisen wih or inerreaion rom he Q ecor analsis. More Eamles We will work hroh more eamles o how he Q-ecor orm o he qasi-eosrohic omea eqaion ma be alied o dianose snoic-scale erical moion boh in class, ilizin he NCA/MMM eal-ime Dianosics ae linked rom he corse websie, and in he second laboraor assinmen o he semeser. Howeer, lease do make se o he NCA/MMM websie oside o class o rher aid or own sd and inerreaion o hese conces! Adanced Inerreaion, Geosrohic Balance, and Alicaion o Fronoenesis Adanced Inerreaion: elaion o Horizonal emerare Gradien o his oin, we hae discssed how he Q ecor ma be comed and/or esimaed. Likewise, we hae shown how is dierence can be sed o iner he direcion and manide o he snoic-scale erical moion. Howeer, we hae e o describe he hsical meanin o he comonens o he Q ecor. ha is he ocs o his secion. Eqaions (19) and (1) ie or eressions or Q 1 and Q, reseciel. Eaminin hese eressions, we see ha boh look like emerare adecion b he horizonal shear o he eosrohic wind. Alernaiel, Q 1 and Q cold be inerreed as bein relaed o he eolion o he horizonal emerare radien. Howeer, i is air o ask wheher hese inerreaions are he bes ossible inerreaions or he Q ecor. o do so, le s rern o he simliied orm o he qasi-eosrohic hermodnamic eqaion osed in (3). Le's make his eqaion een simler: le s sae ha he low is rel eosrohic sch ha he erical eloci erm S ω anishes. his allows s o wrie: (7) Noe ha we hae eanded he adecion erm in (3) ino is comonens in wriin (7). We irs wish o ind / o (7). In so doin, we ms make se o he rodc rle when akin he arial deriaies o he second and hird erms on he le-hand side o (7). Doin so, we obain: he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 11

12 he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 1 (8) Ne, we wish o comme he order o he arial deriaies in he irs and las erms on he le-hand side o (8). Doin so and roin like erms, we obain: (9) he las wo erms on he rih-hand side o (9) can be wrien in ecor orm, reslin in: (3) Howeer, we know ha his erm is relaed o Q 1 in ac, rom (19), we know ha i is eqialen o Q 1 /. Moin his erm o he rih-hand side o (3), we obain: Q D D 1 (31) We now wish o ind / o (7). In so doin, we ms aain make se o he rodc rle when akin he arial deriaies o he second and hird erms on he le-hand side o (7). Doin so, we obain: (3) Ne, we wish o comme he order o he arial deriaies in he irs and hird erms on he le-hand side o (3). Doin so and roin like erms, we obain: (33) he las wo erms on he rih-hand side o (33) can be wrien in ecor orm, reslin in: (34) Howeer, we know ha his erm is relaed o Q in ac, rom (1), we know ha i is eqialen o Q /. Moin his erm o he rih-hand side o (34), we obain:

13 Combinin (31) and (35), we obain: D Q (35) D D Q ˆi ˆ 1 Q j (36) D Eqaion (36) demonsraes ha he Q ecor can be inerreed as bein roorional o he rae o chane o he horizonal emerare radien as orced eclsiel b he eosrohic moion. We will rern o his inerreaion momenaril. Alicaion o Geosrohic Balance Eqaions (1) and (19) sae he deiniions o Q and Q 1, reseciel. he orcins on Q 1 and Q are enirel eosrohic in nare, wheher direcl ( ) or indirecl (, relaed o eooenial heih ia he hdrosaic eqaion). hs, as we discssed in he cone o he qasi-eosrohic omea eqaion, rel eosrohic low is resonsible or dearres rom eosroh (i.e., or aeosrohic low). he reslan aeosrohic circlaion, which is relaed o Q 1 and Q b (11) and (18), works o resore eosrohic and hermal wind balance. We can rher demonsrae he conce o eosrohic low bein resonsible or dearres rom eosroh in he cone o he hermal wind relaionshi. I shold be noed, howeer, ha we will likel no coer his orion o he maerial in-deh in class, nor will o be resonsible or i on qizzes or eams. Firs, we wish o se he hermal wind relaionshi osed in (7) o re-wrie (31) and (35) in erms o he erical shear o he eosrohic wind. Sbsiin (7b) ino (35) and (7a) ino (31), we obain: D D D D Q Q 1 (37a) (37b) For comarison, we now rern o eqaions (1) and (). As we did wih he simliied orm o he qasi-eosrohic hermodnamic eqaion, we wish o consider he secial case where he low is rel eosrohic. his allows s o sae: he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 13

14 he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 14 (38a) (38b) B indin / o (38a) and (38b), we can deelo alernae eressions or he le-hand sides o (37a) and (37b). We make se o he rodc rle and comme he order o ariclar arial deriaies in obainin hese eressions, as we did in eqaions (8) hroh (3) when oerain on he qasi-eosrohic hermodnamic eqaion. In mahemaical orm, D D (39a) D D (39b) I we al he hermal wind relaionshi ien b (7) o re-wrie he second and hird erms o (39a) and (39b), we obain: D D (4a) D D (4b) Becase he dierence o he eosrohic wind or consan is zero, he ollowin saemen is re: (41) We can al (41) o he second erm on he rih-hand sides o (4a) and (4b) o obain: D D (4a)

15 D D (4b) Sbsiin rom (1) and (19), he deiniions o Q1 and Q, we obain: D D D D Q Q 1 (43a) (43b) Comare (43) o (37). he le-hand sides o boh eqaions are eqialen. Howeer, while he rih-hand sides o boh eqaions are o eqialen manide, he are o oosie sin! his means ha he orcin rom he emerare radien and erical wind shear are no in balance. I hermal wind balance were mainained, (37) and (43) wold no be oosie in sin. Insead, becase his sin discreanc eiss, eosrohic orcin manies hroh he Q ecor desros hermal wind balance. he aeosrohic circlaion and accomanin erical moion also manies hroh he Q ecor, as described aboe, works o ose his sin discreanc and hs aem o resore hermal wind balance. Alicaion o Fronoenesis We close b rernin o (36), he relaionshi beween he Q ecor and he rae o chane o he horizonal emerare radien as orced b he eosrohic moion. ecall ha we can analze rons in erms o horizonal emerare radiens. Cold rons are icall ond on he leadin ede o cold air, while warm rons are icall ond on he leadin ede o warm air. Across he ron, here is a lare horizonal radien o emerare. How his radien or, more seciicall, is manide chanes wih ime ies a measre o how he inensi o he ron chanes wih ime. o elore his idea rher, we wan o deelo a relaionshi or he rae o chane o he manide o he horizonal emerare radien: D D ( ) D D (44) We desire o eand he rih-hand side o (44). In doin so, we make se o he ollowin eneral relaionshi: he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 15

16 D D D (45) D n n ( ) n ( ) 1 In (45), is some arbirar ncion and n is some arbirar eonen or ower. Makin se o (45) as we eand he rih-hand side o (44), we obain: D D ( ) 1/ 1 1 D D 1 D D D D D D D D (46) I we sbsie he deiniion o Q, as ien b (31) and (35), (46) becomes: D D 1 ( ) Q Q 1 (47) Or, in ecor noaion, D 1 D ( ) ( Q ) (48) Wha does (48) sini? his eqaion siniies ha he rae o chane o he manide o he horizonal emerare radien b rel eosrohic rocesses, he le-hand side o (48), is roorional o how he horizonal emerare radien ( ) and Q ecors (Q) are oriened wih resec o one anoher. his laer remark abo he orienaion o he wo ecors arises rom he deiniion o he do rodc conained wihin (48). o aciliae inerreaion o (48), i is hell o recall he roeries o he do rodc: For an wo ecors A and B, i A is erendiclar o B, heir do rodc is zero. I A oins in he same direcion as B, heir do rodc is osiie. I A oins in he oosie direcion as B, heir do rodc is neaie. As alied o (48), he rae o chane o he manide o he horizonal emerare radien b rel eosrohic rocesses is zero i he horizonal emerare radien (alwas direced rom he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 16

17 cold oward warm air) and Q ecors are erendiclar o one anoher. I he horizonal emerare radien and Q ecors oin in he same direcion, he rae o chane o he manide o he horizonal emerare radien is osiie. Conersel, i he horizonal emerare radien and Q ecors oin in he oosie direcion, he rae o chane o he manide o he horizonal emerare radien is neaie. Le s al his conce o or Eamle 3, ha o conlen low ino a je sreak, rom earlier in his lecre. he horizonal emerare radien in ha eamle oins rom norh o soh, rom cold oward warm air. Likewise, in ha eamle, we demonsraed ha he Q ecors oin rom norh o soh. hs, in his case, he horizonal emerare radien and Q ecors oin in he same direcion. his means ha he manide o he horizonal emerare radien will row larer wih ime, a ronoeneic siaion. We can conirm his b consider how he eosrohic wind, ien b he sreamlines in Fire 5, blows across he isoherms. o he norhwes, he eosrohic wind blows rom cold oward warm air. Conersel, o he sohwes, he eosrohic wind blows rom warm oward cold air. his aern o eosrohic emerare adecion acs o increase he manide o he horizonal emerare radien o he wes, as we dedced aboe. his eercise can be reeaed or dilen low in he ei reion comin o o a je sreak. In ha case, he Q ecors and horizonal emerare radien oin in oosie direcions, casin he manide o he horizonal emerare radien o become smaller wih ime, a ronolic siaion. o ain eerience wih hese conces, I encorae o o work hroh his eercise on or own and o ask me i o rn ino an roble in so doin. hs, in he qasi-eosrohic ssem, he deelomen and deca o rons can be ealaed b considerin how he Q ecors are oriened wih resec o he isoherms! he dal ili o he Q ecor o ealae boh snoic-scale erical moion (and is associaed aeosrohic low) as well as he deelomen and deca o rons illsraes e anoher owerl adanae o he Q ecor ormlaion oer he normal orm o he qasi-eosrohic omea eqaion! I also illsraes how ascen, clods, and reciiaion are oen ond in reions o ronoenesis, ien he sron relaionshi beween erical moion and ronoenesis manies b he Q ecor. I shold be noed, howeer, ha he aboe ramework onl considers how eosrohic rocesses ac o chane he manide o he horizonal emerare radien. Aeosrohic rocesses ma be and oen are imoran conribors o his as well. Adanced Alicaion: he For Qadran Je Model Earlier, we ealaed he Q-ecors (and, hs, erical moion) or he case o conlen low in he enrance reion o a je sreak wih isoherms oriened arallel o he je. here is Q-ecor conerence in he rih enrance reion o he je sreak, indicain middle roosheric ascen, he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 17

18 and Q-ecor dierence in he le enrance reion o he je sreak, indicain middle roosheric descen. Similar armens can be osed or he ei reion o he je sreak. In ha case, he Q-ecors are oriened rom soh o norh, indicaie o Q-ecor conerence in he le ei reion and Q-ecor dierence in he rih ei reion o he je. his indicaes middle roosheric ascen and descen, reseciel. One adanae o he Q-ecor-based inerreaion o he or qadran je model is is elici consideraion o he hermal radien or, in oher words, he horizonal disribion o emerare. I he isoherms are no oriened arallel o he je sreak, he disribion o ascen and descen will be chaned somewha rom ha seen in he idealized eamle resened herein. he Q-ecor-based inerreaion accons or his direcl; he Q-G omea eqaion-based inerreaion can do so as well i is hermal orcin erm is ealaed. he arcel acceleraionbased inerreaion, howeer, canno do so direcl. he Q-Vecor Form o he Qasi-Geosrohic Omea Eqaion, Pae 18